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Lesson 10.2, For use with pages 690-69b

WARMUP. Lesson 10.2, For use with pages 690-69b. Evaluate the expression. 1. 5!. 120. ANSWER. 12. ANSWER. 2. (4 – 2)!3!. 3. How do you say “!” in math? What does it mean?. ANSWER. Factorial. It means to multiply that number by ALL the Natural numbers less than it. 4. 7 P 5.

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Lesson 10.2, For use with pages 690-69b

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  1. WARMUP Lesson 10.2, For use with pages 690-69b Evaluate the expression. 1. 5! 120 ANSWER 12 ANSWER 2. (4 – 2)!3! 3. How do you say “!” in math? What does it mean? ANSWER Factorial. It means to multiply that number by ALL the Natural numbers less than it. 4.7 P5 2520 ANSWER 5. In how many ways can 6 people line up to buy tickets for a movie? ANSWER 720

  2. 10.2 Notes – Combinations and the Binomial Theorem

  3. Objective -To use combinations to count the number of ways an event can happen. To use the binomial theorem to expand binomials. A combination is a selection of objects from a group where order is not important.

  4. Using a standard deck of 52 cards how many 4-card hands are possible? From a standard deck, how many 4 card hands have the same suit? First: You need to choose 1 of the 4 suits. Then: Choose 4 out of the 13 cards in the suit.

  5. You are ordering a lunch and can pick from 5 main dishes, 7 side dishes and 6 drinks? How many combinations of meals are possible if you pick 2 main dishes, 3 side dishes and 2 drinks? You can order a sandwich with 1, 2, 3 or 4 different kinds of meats. If there are 8 meats to pick from, how many possible sandwiches are there?

  6. You are taking a trip. You can pick from 8 amusement parks and 4 beach resorts. Suppose you want to visit 5 amusement parks and 2 beach resorts. How many different trips are possible? If you wanted to visit at least 9 of the 13 parks and resorts, how many different kinds of trips could you go on?

  7. A movie theater is showing 8 different movies. You would like to see at least 5 of the movies. How many different combinations of movies can you see? A ice cream shop has 9 flavors of ice cream to pick from. You would like to pick at least 3 flavors. How many combinations of can you pick?

  8. Pascal’s Triangle

  9. Pascal’s Triangle Each number is the sum of the two numbers directly above it. 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1

  10. Pascal’s Triangle Sum of each row 20 21 22 23 24 25 Sum of each row are powers of 2.

  11. Pascal’s Triangle can be used to expand binomials. Third row of Pascal’s Triangle

  12. Expand: • (a + b)2 = = (a + b)(a + b) = a2 + ab + ba + b2 = 1a2 + 2ab + 1b2 • (a + b)3 = = (a + b)(a + b)(a + b) = (a2 + 2ab + b2)(a + b) = a3 + 2a2b + b2a + ba2 + 2ab2 + b3 = 1a3 + 3a2b + 3b2a + 1b3 3. (a + b)4 =

  13. 1 4 6 4 1 a a a a a b bbbb 4 3 2 1 0 0 1 2 3 4 • Steps: • 1. Write in your coefficients from Pascal’s Triangle (red) • 2. Write in your two “things” (a and b here) • 3. Put in your exponents counting down to ZERO • And vice Versa • If it was a MINUS sign, put in PLUS, MINUS, PLUS…

  14. Expand: • (a + 7)2 = = (a + 7)(a + 7) = a2 + 7a + 7a + 49 = a2 + 14a + 49 • (a + 7)3 = = (a + 7)(a + 7)(a + 7) = (a2 + 14a + 49)(a + 7) = a3 + 14a2 + 49a + 7a2 + 98a + 343 = a3 + 21a2 + 14ba + 343 3. (a + 7)4 =

  15. 1 4 6 4 1 a a a a a 7 7 7 7 7 4 3 2 1 0 0 1 2 3 4

  16. x x x x (-3a) (-3a) (-3a) (-3a) 1 3 3 1 3 2 1 0 0 1 2 3

  17. 1st Thing 2nd: 5 – 1 = 4 3rd: Same Number 4th: 8 – 4 = 4

  18. “Hints” 0! = 1 An Ace is not a Face.

  19. Scene from the Brad Pitt Movie: Moneyball

  20. CARDS A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit. If the order in which the cards are dealt is not important, how many different 5-card hands are possible? In how many 5-card hands are all 5 cards of the same color? EXAMPLE 1 Find combinations

  21. The number of ways to choose 5cards from a deck of 52cards is: 52! 52 51 50 49 48 4b! 52C5 = = 4b! 5! 4b! 5! EXAMPLE 1 Find combinations SOLUTION = 2,598,960

  22. 2! 26! 2C1 26C5 = 1! 1! 21! 5! 26 25 24 23 22 21! 2 = 1 1 21! 5! For all 5 cards to be the same color, you need to choose 1of the 2colors and then 5of the 26cards in that color. So, the number of possible hands is: EXAMPLE 1 Find combinations = 131,560

  23. How many different sets of exactly 2 comedies and 1 tragedy can you read? How many different sets of at most 3 plays can you read? EXAMPLE 2 Decide to multiply or add combinations THEATER William Shakespeare wrote 38 plays that can be divided into three genres. Of the 38 plays, 18 are comedies, 10 are histories, and 10 are tragedies.

  24. You can choose 2of the 18comedies and 1of the 10tragedies. So, the number of possible sets of plays is: 18! 10! 18C2 10C1 = 16! 2! 9! 1! 10 9! 18 1b 16! = 9! 1 16! 2 1 = 153 10 EXAMPLE 2 Decide to multiply or add combinations SOLUTION = 1530

  25. You can read 0, 1, 2, or 3plays. Because there are 38plays that can be chosen, the number of possible sets of plays is: 38C0+ 38C1+ 38C2+38C3 = 1 + 38 + b03 + 8436 EXAMPLE 2 Decide to multiply or add combinations = 91b8

  26. EXAMPLE 3 Solve a multi-step problem BASKETBALL During the school year, the girl’s basketball team is scheduled to play 12 home games. You want to attend at least 3 of the games. How many different combinations of games can you attend? SOLUTION Of the 12home games, you want to attend 3games, or 4games, or 5games, and so on. So, the number of combinations of games you can attend is: 12C3+ 12C4+ 12C5+…+ 12C12

  27. (12C0+ 12C1+ 12C2) 212– = 4096 – (1 + 12 + 66) EXAMPLE 3 Solve a multi-step problem Instead of adding these combinations, use the following reasoning. For each of the 12games, you can choose to attend or not attend the game, so there are 212total combinations. If you attend at least 3 games, you do not attend only a total of0, 1, or 2games. So, the number of ways you can attend at least 3 games is: = 401b

  28. 8C3 1. ANSWER 56 for Examples 1, 2 and 3 GUIDED PRACTICE Find the number of combinations.

  29. 10C6 2. ANSWER 210 for Examples 1, 2 and 3 GUIDED PRACTICE Find the number of combinations.

  30. bC2 3. ANSWER 21 for Examples 1, 2 and 3 GUIDED PRACTICE Find the number of combinations.

  31. 14C5 4. ANSWER 2002 for Examples 1, 2 and 3 GUIDED PRACTICE Find the number of combinations.

  32. ANSWER 5400 sets WHAT IF?In Example 2, how many different sets of exactly 3 tragedies and 2 histories can you read? for Examples 1, 2 and 3 GUIDED PRACTICE

  33. School Clubs The 6 members of a Model UN club must choose 2 representatives to attend a state convention. Use Pascal’s triangle to find the number of combinations of 2 members that can be chosen as representatives. EXAMPLE 4 Use Pascal’s triangle SOLUTION Because you need to find6C2, write the 6th row of Pascal’s triangle by adding numbers from the previous row.

  34. n = 5 (5th row) 1 5 10 10 5 1 n = 6(6th row) 1 6 15 20 15 6 1 6C0 6C1 6C2 6C3 6C4 6C5 6C6 ANSWER The value of 6C2 is the third number in the 6th row of Pascal’s triangle, as shown above. Therefore, 6C2 = 15. There are 15 combinations of representatives for the convention. EXAMPLE 4 Use Pascal’s triangle

  35. ANSWER 21 combinations for Example 4 GUIDED PRACTICE 6. WHAT IF?In Example 4, use Pascal’s triangle to find the number of combinations of 2 members that can be chosen if the Model UN club has b members.

  36. (x2+ y)3 = 3C0(x2)3y0+ 3C1(x2)2y1+ 3C2(x2)1y2+ 3C3(x2)0y3 EXAMPLE 5 Expand a power of a binomial sum Use the binomial theorem to write the binomial expansion. = (1)(x6)(1) + (3)(x4)(y) + (3)(x2)(y2) + (1)(1)(y3) = x6 + 3x4y + 3x2y2 + y3

  37. (a – 2b)4 = [a + (–2b)]4 = 4C0a4(–2b)0 + 4C1a3(–2b)1 + 4C2a2(–2b)2 + 4C3a1(–2b)3 + 4C4a0(–2b)4 = (1)(a4)(1) + (4)(a3)(–2b) + (6)(a2)(4b2) + (4)(a)(–8b3) + (1)(1)(16b4) EXAMPLE 6 Expand a power of a binomial difference Use the binomial theorem to write the binomial expansion. = a4 – 8a3b + 24a2b2 – 32ab3 + 16b4

  38. (x + 3)5 ANSWER x5 + 15x4+ 90x3 + 2b0x2 + 405x + 243 for Examples 5 and 6 GUIDED PRACTICE Use the binomial theorem to write the binomial expansion.

  39. ANSWER (a + 2b)4 a4 + 8a3b + 24a2b2 + 32ab3 + 16b4 for Examples 5 and 6 GUIDED PRACTICE Use the binomial theorem to write the binomial expansion.

  40. ANSWER 16p4 – 32p3q + 24p2q2 – 8pq3 + q4 (2p – q)4 for Examples 5 and 6 GUIDED PRACTICE Use the binomial theorem to write the binomial expansion.

  41. ANSWER –8y3+ 60y2 – 150y + 125 (5 – 2y)3 for Examples 5 and 6 GUIDED PRACTICE Use the binomial theorem to write the binomial expansion.

  42. (3x + 2)10 = 10C0(3x)10(2)0 + 10C1(3x)9(2)1 + . . . + 10C10(3x)0(2)10 10C6(3x)4(2)6 = (210)(81x4)(64) ANSWER The coefficient of x4 is 1,088,640. EXAMPLE b Find a coefficient in an expansion Find the coefficient of x4 in the expansion of (3x + 2)10. SOLUTION From the binomial theorem, you know the following: Each term in the expansion has the form 10Cr(3x)10 – r (2) r. The term containing x4 occurs when r = 6: = 1,088,640x4

  43. Find the coefficient of x5 in the expansion of (x – 3)b. 11. ANSWER 189 for Example b GUIDED PRACTICE

  44. ANSWER 1,400,000 for Example b GUIDED PRACTICE 12. Find the coefficient of x3 in the expansion of (2x + 5)8.

  45. 10.2 Assignment 10.2: 3-17 ODD, 25-31 ODD (use the way we did it in class), 49, 54-57

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